Pedal curve

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Limaçon — pedal curve of a circle
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Limaçon — pedal curve of a circle

In the differential geometry of curves, a pedal curve is a curve derived by construction from a given curve (as is, for example, the involute).

Take a curve and a fixed point P (called the pedal point). On any line T is a unique point X which is either P or forms with P a line perpendicular to T. The pedal curve is the set of all X for which T is a tangent of the curve.

The pedal "curve" may be disconnected; indeed, for a polygon, it is simply isolated points.

Analytically, if P is the pedal point and c a parametrisation of the curve then

t\mapsto c(t)+{\langle c'(t),P-c(t)\rangle\over|c'(t)|^2} c'(t)

parametrises the pedal curve (disregarding points where c' is zero or undefined).

The contrapedal curve is the set of all X for which T is perpendicular to the curve.

t\mapsto P-{\langle c'(t),P-c(t)\rangle\over|c'(t)|^2} c'(t)

With the same pedal point, it happens to be the pedal curve of the evolute.

In the plane, for every point X other than P there is a unique line through X perpendicular to XP. The negative pedal curve is the envelope of the lines for which X lies on the given curve. The negative pedal curve of a pedal curve with the same pedal point is the original curve.

given
curve		 pedal
point		 pedal
curve		 contrapedal
curve
line any point parallel line
circle on circumference cardioid —
parabola on axis conchoid of de Sluze —
parabola on tangent
of vertex		 ophiuride —
parabola focus line —
other conic section focus circle —
logarithmic spiral pole congruent log spiral congruent log spiral
epicycloid
hypocycloid		 center rose rose
involute of circle center of circle Archimedean spiral the circle

[edit] Example

Pedal curves of unit circle:

c(t) = (cos(t),sin(t))
c'(t) = ( − sin(t),cos(t))   and   | c'(t) | = 1
{\langle c'(t),(x,y)-c(t)\rangle\over|c'(t)|^2}=y\cos(t)-x\sin(t)

thus, the pedal curve with pedal point (x,y) is:

(cos(t) − ycos(t)sin(t) + xsin(t)2,sin(t) − xsin(t)cos(t) + ycos(t)2)

If the pedal point is at the center (i.e. (0,0)), the circle is its own pedal curve. If the pedal point is (1,0) the pedal curve is

(cos(t) + sin(t)2,sin(t) − sin(t)cos(t)) = (1,0) + (1 − cos(t))c(t)

i.e. a pedal point on the circumference gives a cardioid.

[edit] External links

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